In a hot strip mill, the precision of product thickness is very important for the quality of the product. Therefore, thickness control must be achieved. In order to accommodate hot strip mill plants of reduced size and improve their efficiency, it has been considered to introduce a coil box to thereby shorten the overall length of the line and to establish a F.sub.o stand to thereby reduce unit consumption of the roll and prevent the temperature drop.
The set-up of a finishing mill includes estimating the rolling force P.sub.i.sup.CAL and calculating the roll gap S.sub.i of each of the mill stands on the basis of the roll force estimation, where i=1, 2, . . . , n, with which gaps and speeds, etc., of the mill stands are set-up in an actual plant. In a case where the estimated rolling force P.sub.i.sup.CAL of the mill stand is not coincident with the actual value P.sub.i.sup.ACT, the accuracy of thickness of a product is lowered. In order to solve this problem, i.e., to make an actual product thickness h.sub.F.sup.ACT coincident with objective product thickness h.sub.F.sup.AIM, it is necessary to make the estimated rolling forces P.sub.i.sup.CAL of the mill stand coincident with actual rolling force P.sub.i.sup.ACT.
In general, the rolling force P.sub.i.sup.CAL can be represented by the following equation when there is no tension: EQU P.sub.i.sup.CAL =k.sub.mi .multidot.L.sub.d .multidot.B.multidot.Q.sub.pi . . . ( 1)
where k.sub.mi is a mean resistance to deformation of a workpiece in an i-th stand F.sub.i, L.sub.d is length of arc of contact, B is a width of sheet workpiece and Q.sub.pi is a rolling force function of the i-th stand F.sub.i.
As is clear from the equation (1), the accuracy of the calculated rolling force P.sub.i.sup.CAL is determined by the mean resistance to deformation k.sub.mi and the rolling force function Q.sub.pi. The rolling force function Q.sub.pi represents the geometrical characteristic of rolling and the mean resistance to deformation k.sub.mi represents a physical characteristic of the workpiece. That is, a model of mean resistance of deformation is supplied with a mean temperature T.sub.i of the workpiece under rolling, strain.epsilon..sub.i and strain rate .epsilon..sub.i as input parameters and the mean resistance to deformation which is generally influenced largely by the chemical components of steel material. Model equation includes a coefficient table having sections classified with every equivalent equation of component or type of steel, so that a difference in type of steel can be represented.
A basic equation of the mean resistance to deformation model is represented as follows: ##EQU1## where ##EQU2## are static components of mean resistance to deformation and ##EQU3## are kinematic components. The static components ##EQU4## are a function of the strain .epsilon..sub.i and the temperature T.sub.i, and independent of the strain rate .epsilon..sub.i. The kinematic components ##EQU5## are functions of the strain rate .epsilon..sub.i and the temperature T.sub.i and independent from the strain .epsilon..sub.i Either the multiplicative equation (2) or the additive equation (3) is on the stress-strain curve of a metal material. The static component ##EQU6## representing the stress-strain curve, is generally represented by the n-th power law of hardening according to work hardening or softening depending upon temperature, strain rate and/or kind of steel, as follows: EQU K.sub.s =C.multidot..epsilon..sup.n ( 4)
where C and n are constants depending upon the kind of steel and the temperature. This equation can represent only the work hardening. In order to represent the work softening, it is sufficient to add a term of difference in strain .epsilon. to the equation (4). In this description, since it is based on the n-th power law even considering the work softening, there is no substantial difference.
The resistance to deformation of a finishing stage of a hot strip mill is generally determined by a transformation of the equation (4) into a two-dimensional mean resistance to deformation. That is, EQU k.sub.s =1.15 .multidot.C.multidot..epsilon..sup.n ( 5)
For example, the strain .epsilon..sub.i in i-th stand F.sub.i is obtained from the reduction r.sub.i of that stand. The strain is defined in various manner. When defined by Sims' definition which is widely used, it is expressed by EQU .epsilon..sub.i =-ln(1-r.sub.i) (6)
and the reduction r.sub.i of the i-th stand F.sub.i is expressed by EQU r.sub.i =(H.sub.i -h.sub.i)/H.sub.i ( 7)
where H.sub.i is the thickness of a workpiece entering into the i-th stand F.sub.i and h.sub.i is the delivery thickness, i.e., the thickness thereof leaving the stand F.sub.i.
In this manner, when the reduction ri of each stand F.sub.i is used, the resistance to deformation k.sub.s is calculated from the strain .epsilon..sub.i for every stand F.sub.i as shown in FIG. 2. This does not cause any severe problem in the case of a hot strip mill having no soaking facility such as a coil box. It is capable of maintaining a constant temperature on the entrance side of the finishing mill since the strain rate with respect to the roughing mill is sufficiently large. However, for a hot strip mill having such a facility, it is impossible to express the thermal effect, and, thus the accuracy of the calculation is not sufficient.
This problem is caused by the assumption that the material enters into the i-th stand F.sub.i with its strain .epsilon..sub.(i-1) given by a preceding stand F.sub.(i-1) being fully recovered, since the strain .epsilon..sub.i in the finishing mill depends basically upon the thickness of the workpiece on the entrance side of each stand and the delivery thickness thereof. That is, in the mill having a coil box on its entrance side, the material wound in the coil box does not experience any temperature drop due to inter-layer radiation, resulting in a metal structure similar to that annealed by soaking effect and which is supplied to the finishing stand. In the conventional method in which material strain in each stand is used, it is impossible to reflect the influences of the temperature of the material on the side of the coil box facing the entrance side of the finishing mill and the metallurgical structure thereof to the deformation resistance model. Therefore, the accuracy of estimation of the rolling force of the finishing mill becomes insufficient, resulting in a final product thickness which is inaccurately controlled.
As mentioned, in the past, the deformation resistance of the finishing stand is estimated on the basis of the reduction r.sub.i and the strain .epsilon..sub.i without considering the thermal effect of the coil box arranged on the entrance side of the finishing stand. Therefore, the soaking varies with the sheet thickness HR (referred to as transfer bar thickness) during winding in the coil box. It is impossible to express the effect thereof to the rolling force of the finishing stands and it is impossible to handle features of the material of the workpiece to be rolled easily, because the structure thereof has changed in the coil box to one similar to that annealed.
In a mill having a coil box, the effect of the coil box is not reflected effectively to the setting of the finishing mill by calculating the estimated rolling force P.sub.i.sup.CAL in each Of the respective finishing stands with respect to a state (temperature, deformation and structure) of the workpiece wound in the coil box. Since, in the prior art, the rolling forces P.sub.i.sup.CAL in each of the respective stands are calculated on the basis of the reduction r.sub.i of the stand, and the gap S.sub.i at the stand F.sub.i is set on the basis of the result, P.sub.i.sup.CAL, of the calculation, the accuracy of thickness of the final product is degraded.